■'SOf^E FLOW PROBLEMS
A THESIS ON
I N IIYDROivG rETICS AND HYDROBY W iK IC S
By
H*L« S e t h i
Department o f Mathematics I n d i a n I n s t i t u t e o f Technology
New D el h i
Submitted to the I n d i a n I n s t i t u t e o f T e ch n ol og y , New D e lh i f o r the award o f the Degree o f Doctor o f P h i l o s o p h y
i n Mathematics
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1968C E k T I F I C A T S
This is to c e r t i f y t h a t the t h e s i s e n t i t l e d "Some Flow Problems i n H y d r on a g n et ie s and Hydrodynamics" which i s b e in g
su bm it te d by J h r i Harbsns L a i S a t h i to the Indian. I n s t i t u t e o f T e c h n o lo g y , J el h i f o r the award o f the Degree o f I h i l o s o p h y
( K a t h e n a t i c s ) i j a r e c o r d o f bona f i d e r e s e a r c h work. Re has
worked under g ui d ar c e and s u p e r v i s i o n f o r the l a s t t h r e e y e a r s and t h r e e months.
The t h e s i s has 'reached the standard f u l f i l l i n g the r e q u i r e ments o f the rcgi il e t i o n s r e l a t i n g to the d e g r e e . The r e s u l t s o b t a i n e d i n th is t h e s i s have not been sub m it te d t o any o t h e r u n i v e r s i t y o r i n s t i t u t e f o r t h e award o f any d e g r e e or diploma.
( F,I.P, S i n g h " )
Department o f
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a t h e ^ a t i c s I n d i a n I n s t i t u t e o/' Techno Kauz " h a s , ?Je».v 3 e l h i - 2 9 .I am e x t r e mely grc t e f u l to Dr, i.i«P« Singh, i..«Sc.
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Ph» 3. , Depsrtiaent. o f Lt t h e m c t i c s , I n d i a n I n s t i t u t e o f T e c h n o lo g y , Delhi f o r h i s v a l u a b l e guidance throughout t h i s r es ea rch work, Iexp res s my deep sense o f g r a t i t u d e to P r o f e s s o r M<(.* J a i n , *,.».. *«, D „ P h i l * , D* S c , , Her d o f the Department o f Mathematics, I n d i a n I n s t i t u t e o f Te ch n ol og y, De lh i f o r h i s co n st a n t encouragement, i n s p i r a t i o n and u s e f u l s u g g e s t i o n s ,
Ly thanlcs £ re due to P r o f e s s o r R. N, Dagra, D i r e c t o r ,
I n d i e n I n s t i t u t e o f Te chn olo gy , Delhi f o r a l l the f a c i l i t i e s I r e c e i v e d during the c o u r s e o f t h i s work*
I s h a l l f a i l i n my duty i f I do not thank Dr. G„P. Bhutani, Mr. k,M, Chav/la & nd o t h e r c o l l e a g u e s i n the Department f o r f r u i t f u .
d i s c u s s i o n s . I wish to thank Mr* Joseph Kurian, Mr * D» i>u bra mania!
and fc iss . Juneja o f the Computer C e n t r e , Incii?r. I n s t i t u t e o f T e c h n o lo g y , Delhi f o r t h e i r kind h e l p and c o - o p e r a t i o n i n the comp uta tio n work done f o r t h i s t h e s i s ,
I put on r e c o r d the a s s i s t a n c e I r e c e i v e d from Mr. H,R,Thakr i n drawing the graphs* I a l s o thank Mr* Dev Raj J o s h i and f i r , N.I Sachdev f o r t h e i r commendable work i n t y p i n g t h i s manuscript*
L a s t but not the l e a s t , I am de ep ly i n d e b t e d to my w i f e , Rirst Pushpa S a t h i , who has stood by me through a l l s t r e s s e s and
s trc i n s ,
l-H*. Scu&zi.
( H.L* S a t h i ) Department o f Mathema;
I n d i a n I n s t i t u t e o f
f
Hauz Khas, New Delhi/S Y N O P S I S
The importance o f the e f f e c t s o f r o t a t i o r j , s u c t i o n , s l i p and magnetic f 1 e l a on the f l u i d f l o w ana he at t r a n s f e r problems i s well_kno»vn.
R o t a t i o n pia; s a s i g n i f i c a n t r o l e i n s e v e r a l im po rt an t phenomenon i n cos.r-ical f l u i d dynamics. Simili. r l y , a g r e a t deal o f m e t e o r o l o g y ue.-cr.os upon the dynamics o f a r e v o l v i n g f l u i d .
The l a r g e s c a l e : net moderate motions o f the atmosphere a r e g r e a t l y a f f e c t e d by the v o r t i c i t y o f e a r t h ’ s r o t a t i o n . I n the c a s e o f an
i n f i n i t e f l u i d r o t a t i n g as a r i g i d body about an a * i £ , the amount o f energy possessed by the l i q u i d i s i n f i n i t e and i t i s o f g r e a t i n t e r e s t to see ooT" small di s tu r b a n c e s p r o p a g a t e i n such a l i q u i d .
The e f f e c t s o f s u c t i o n are im portant i n c o n t r o l l i n g the boundary l_.yer anJ thus a v o i d i n g s e p a r a t i o n which f u r t h e r r e s u l t s i n the r e d u c t i o n o f s k i n f r i c t i o n on the hedy. I t a l s o causes d e la y i n t r a n s i t i o n
fro n
the .lam ina r to the t u r b u l e n t f l o w .The develo.uent o f r o c k e t s whether f o r b a l l i s t i c m i s s i l e s o r de si g ne d to pro.be i n t o the o u t e r space, has g i v e n
t.
new impetusto the study o f r a r e f i e d gas dynamics. I t has been seen t h a t f o r l a r ^ e Knudsen number, the gas i s c o n s i d e r e d to be i n r a r e f i e d s t a t e and k i n e t i c theory approach i s more a p p r o p r i a t e . But f o r small Xnudsen number, the f l o w o f gas i s i n the s o - c a l l e d ’ s l i p ' r e g i o n and the use o f continuum t h e o r y w it h m o d i f i e d boundary c o n d i t i o n s .
a l l o w i n g f o r a s l i p a t the s u r f a c e g i v e s good aooroximt'te r e s u l t s . R e c e n t l y , i t hr s been r e c o g n i s e d t h a t the e l e c t r o m a g n e t i c
i n t e r a c t i o n on the f l u i d f l o w l e s d s to
&
much more valued phenomenon.Thus i t has been the usual p r a c t i c e to extend the problems o f
hydrodynamics to hvciromagnetics, a new but f a s t d e v e l o p i n g s c i e n c e as w e l l as t e c h n o l o g y . This l a t t e r d i s c i p l i n e has made i t s e l f f e l t because o f i t s ovn m e r i t s and has more than proved i t s worth i n the problems o f m i s s i l e and apace dynamics, p r o p u l s i o n s and communication
The purpose o f t h i s t h e s i s i s to i n v e s t i g a t e some problems i n hy drom agnetijs und hydrodynamics r e g a r d i n g f l o w p a t t e r n s , v i s c o u s r e s i s t a n c e a t the w a l l s and he at p r o p a g a t i o n i n t h e r e g i o n under c o n s i d e r a t i o n . I n a l l , i t comprises e i g h t c h a p t e r s . Chapter I i s i n t r o d u c t i o n to the t h e s i s , f o l l o w e d by rem ai nin g se ve n ch ap te rs which a r e the r e s u l t o f t h e o r e t i c a l i n v e s t i g a t i o n s m the ’ s l i p ’
as w e l l as n o - s l i p r e g i o n s . An e x a c t s o l u t i o n o f a r o t a t i n g f l o w i s o b t a i n e d i n ch ap te r I I . The r e s u l t s o f e l e c t r o m a g n e t i c m t e r a c t i o on the f l u i d f l o w a r e r e p o r t e d i n the remaining c h a p t e r s I I I to V I I I . S u c t io n e f f e c t s a re s t u d i e d i n ch ap ter s I I I to V I I .
I n c h a p t e r I which i s o f i n t r o d u c t o r y natu re, a c r i t i c a l survey o f the r e l e v a n t l i t e r a t u r e i s p r e s e n t e d . The fundamental equ ati on s g o v e r n i n g the f l u i d f l o v ; und h e a t t r a n s f e r phenomenon a r e a l s o i n c l u d e d .
I n c h a p t e r I I , a study i s made o f u n s t e a d y f l o w engendered i n a v i s c o u s , i n ' ' c o m o r e s c i b l e , r o t a t i n g f l u i d by an i n f i n i t e f l a t p l a t e s e t suddenly i n .action i n i t s own p la n e . The e x a c t s o l u t i o n o f the problem i s o b t a in e d by the method o f L a p l a c e Transforms. I t i s shown
t h a t the r o t a t i o n induces l a t e r a l motion and i t a l s o g e n e r a t e s v o r t i c I t i s f u r t h e r fo^rci th at i t tends to oppose the p e n e t r a t i o n o f
l o n g i t u d i n a l f l o w . The e f f e c t o f r o t a t i o n on drag i s o b t a i n e d and s e v e r a l l i m i t i n g cases a r e deduced from the s o l u t i o n .
Chapter I I I d e a l s w it h the e f f e c t s o f C o r i o l i s f o r c e s on P„ayleigh problem i n hy dro magnetics. As i n c h a p t e r I I , i t i s seen
t h a t the r o t a t i o n induces l a t e r a l motion. For small r o t a t i o n , i t i s found t h a t the l o n g i t u d i n a l components o f v e l o c i t y , induced magn eti c f i e l d and she;, r s t r e s s remain the same as i n the non
r o t a t i n g ca s e . The l a t e r a l s t r e s s , however, i s se en to d e v e lo p l i n e a r l y with time which g i v e s an erroneous i m p r e s s i o n about the
n o n - e x i s t e n c e o f st e a d y s o l u t i o n . But t h i s i s m i s l e a d i n g and can be e x p l a i n e d from the f a c t t h a t the e x p r e s s i o n o b t a i n e d f o r l a t e r a l shear s t r e s s i s merely the l e a d i n g term o f t h e l i m i t i n g s o l u t i o n o b t a i n e d by c o n s i d e r i n g the angular v e l o c i t y n e g l i g i b l y small i n the s o l u t i o n f o r g e n e r a l r o t a t i o n . Anot her e f f e c t o f r o t a t : i s the g e n e r a t i o n o f v o r t i c i t y as i n t h e non-magnetic c a s e .
The e f f e c t s o f u ni for m s u c t i o n on R a y l e i g h problem i n hy dro- magnetics a r e s t u d i e d i n c h a p t e r IV. The problem i s s o l v e d by the method o f L a p la c e Transforms. For a f i n i t e l y c o n d u c t in g
p l a t e , s o l u t i o n s v a l i d f o r small as w e l l as l a r g e time a r e o b t a in e d f o r magnetic P r . n d t l number u n i t y o r c l o s e to u n i t y . I t i s found t h a t the s h e a r i n g s t r e s s on the p l a t e d e c r e a s e s from i n f i n i t y and tends a s y m p t o t i c a l l y to a f i n i t e v a l u e f o r l a r g e t im e. Thie
l i m i t i n g v a l u e co:.cs out to be g r e a t e r than the c o r r e s p o n d i n g l i m i t i n g v a l u e o.-.-t; i n e d f o r a p e r f e c t l y co nd u cti ng p l a t e i n the absence o f s u c t io n . Furt her, an ex a c t s o l u t i o n i s o b t a i n e d f o r a
no n-conducting >late i n case o f magnetic p r a n d t l number u n i t y and c l o s e to u n i t y . R e s u l t s f o r t h e shear s t r e s s a r e a l s o a n a ly z e d .
I n c h a p t e r V, a study i s made o f the hy dromagnetic f l u c t u a t i n g f l o w p a s t a f l a t p l a t e w it h v a r i a b l e s u c t i o n . I t i s assumed t h a t the s u c t i o n v e l o c i t y normal to th e p l a t e as w e l l as the f r e e stream v e l o c i t y f l u c t u a t e about a mean co n st a n t i n magnitude (b u t not i n d i r e c t i o n ) and t h a t t h e A l f v e n v e l o c i t y i s l e s s than the mean s u c t i o n v e l o c i t y . M l the q u a n t i t i e s o f aerodynamical i n t e r e s t such as v e l o c i t y d i s t r i b u t i o n , s k i n - f r i c t i o n , tem perature f i e l d and he at f l u x upto two harmonic f l u c t u a t i o n s a r e e v a l u a t e d . I t turns out that the amp litude o f f i r s t harmonic f l u c t u a t i o n s m the
s k i n - f r i c t i o n i n c r e a s e s w i th the i n c r e a s e i n the f r e q u e n c y o f f l u c t u a t i o n s , w h i l e t h a t o f second harmonic f l u c t u a t i o n s tends t o a f i n i t e l i m i t . Furt her, f o r s n a i l f r e q u e n c i e s and l a r g e v a l u e s o f s u c t i o n parameter, the phase o f the f i r s t harmonic f l u c t u a t i o n s may be n e g a t i v e but i t u l t i m a t e l y tends to vr/4 a t v e r y high
f r e q u e n c i e s . ,\lso, the phase o f the second
h
.nvonic f l u c t u a t i o n s drops to z e r o v/..or fre qu en c y i s l a r g e . I t i s a l s o seen t h a t f o r the ca s e o f no h e a t t r a n s f e r , t h e r e i s no e f f e c t on the w a l l temperature a t h i g h f r e q u e n c i e s , i . e . , i t remains th e same as i n the non-magnetic case o r m the absence o f v a r i a b l e s u c t i o n . But i n case o f h e a t t r a n s f e r , the amp litude o f t h e f i r s t harmonicf l u c t u a t i o n s i n heat f l u x tends t o a f i n i t e l i m i t and t h e phtse d r o r>
to z e r o . The amp litude o f second harmonic f l u c t u a t i o n s r i s e s w it h f r e q u e n c y and f o r v e r y high f r e q u e n c i e s , the p h ts e l e a d tends to
I n c h a p t e r V I , u s i n g the continuum t h e o r y approach, the
e f f e c t s o f ’ s l i p ’ tire c o n s i d e r e d on the hydro:nagnetic f l u c t u a t i n g f l o w p a s t a porous f l a t p l a t e w i t h o s c i l l a t o r y s u c t i o n . As i n th e e a r l i e r i n v e s t i g a t i o n i n c h a p t e r V, d i s t r i b u t i o n s f o r v e l o c i t y ,
s k i n - f r i c t i o n and tempe rature f i e l d s are c a l c u l a t e d . The c a s e o f f l o v p a s t a f l a t p l i t e i n the n o - s l i p r e g i o n i s deduced as a p a r t i c u l a r c a s e and t h e r e s u l t s i n the two r e g i o n s compared. I t turns uut t h a t i n both ’ s l i p ' as w e l l as n o - s l i p r e g i o n s , the p e r i o d i c s u c t i o n
v e l o c i t y induces more than one harmonic f l u c t u a t i o n s i n the f l o w and te mperature f i e l d s . A ls o f o r both the r e g i o n s , t h e amplitude o f f i r s t harmonic f l u c t u a t i o n i n s k i n - f r i c t i o n i n c r e a s e s w i th t h e
i n c r e a s e i n t h e v ^ l u e o f e i t h e r o f th e magnetic o r s u c t i o n para
m et ers , though i t cannot ex ceed a f i n i t e l i m i t i n the ' s l i p ' f l o w r eg im e. Further, i t i s seen t h a t the amplitude o f second harmonic f l u c t u a t i o n s i n s k i n - f r i c t i o n d e c r e a s e s w it h i n c r e a s e i n the
magnetic pararaeter i n the ' s l i p ' f l o w r e g i o n w h i l e i t i n c r e a s e s i n the n o - s l i p r e g i o n . For v e r y l a r g e v a l u e s o f fre qu en c y i n th e
’ s l i p ' f l o w r e g i o n the phase o f f i r s t harmonic f l u c t u a t i o n s i n s k i n - f r i c t i o n drops t o z e r o — a r e s u l t c o n t r a r y to the one f o r the n o - s l i p r e g i o n . Further f o r both r e g i o n s , second harmonic f l u c t u a t i o i n the s k i n - f r i c t i c n always have a phase l a g whatev er be t h e
fre qu en c y . I t a l s o turns ou t t h a t because o f the f l u c t u a t i n g s u c t i o n even f o r l a r g e v a l u e s o f the r a r e f a c t i o n par ame te r, the f l u c t u a t i o n s i n the temperature f i e l d p e r s i s t which r e s u l t i s q u i t e c o n t r a r y to the one f o r t h e case o f un iform s u c t i o n .
Chapter V I I concefls w it h a more g e n e a r l problem o f hydro- magnetic f l o w past a porous f l a t p l a t e when a time dependent s u c t i o n
i s a p p l i e d a t the w a l l . Two ca se s o f t h i s time dependence are c o n s i d e r e d : ( i ) the s u c t i o n may v a r y p e r i o d i c a l l y , ( i i ) the
s u c t i o n as w e l l as the f r e e stream v e l o c i t y may v a r y s l o w l y but a r e o t h e r w i s e a r b i t r a r y f u n c t i o n s o f time . I t turns uut t h a t t h e s u c t i o n , i n g e n e r a l , induces an unsteady f l o w p a r a l l e l to che w a l l .
Furt her, the p e r i o d i c s u c t i o n changes the shape o f th e mean p r o f i l e . The e f f e c t magnetic f i e l d i s t h a t the mean v ^ l u e o f t h e shear s t r e s s o n . t h e p l a t e i s a l t e r e d - a r e s u l t c o n t r a r y to t h e one o b t a i n e d f o r the non-magnetic c a s e . But when the f r e e stream v e l o c i t y a l s o lias a p e r i o d i c component which o s c i l l a t e s w i t h the s u c t i o n f r e q u e n c y the mean f l o w i s f u r t h e r d i s t o r t e d . For s l o w l y v a r y i n g but o t h e r w i s e a r b i t r a r y s u c t i o n and f r e e stream v e l o c i t y , a s o l u t i o n i s o b t a i n e d
i n terms o f t h e d e r i v a t i v e s o f the s u c t i o n and f r e e stream v e l o c i t i e s . The t h e s i s ends w i t h c h a p te r V I I I wnieh <?eals w i th h y d r o -
mag.netic, f l u c t u a t i n g ' , a x i s - s y m m e t r i c f l o w near t h e s t a g n a t i o n p o i n t r e g i o n . I t i s assumed t h a t the main-stream o u t s i d e t h e boundary l a y e r r e g i o n f l u c t u a t e s i n magnitude (b u t not i n d i r e c t i o n ) about
a steady mean. Using Runge-Kutta method, a n u m e r i c a ll y e x a c t s o l u t i o n f o r the mean f l o w as w e l l as f o r the f l u c t u a t i n g p a r t o f th e f l o w a t small f r e q u e n c i e s i s o b t a i n e d . For l a r g e f r e q u e n c i e s , an approximate sdbition i s d e v e l o p e d f o r th e o s c i l l a t o r y p a r t . The ’ c r i t i c a l frequenc a t which the two s o l u t i o n s o v e r l a p i s a l s o c a l c u l a t e d f o r d i f f e r e n t v a l u e s o f t h e magnetic parameter. I t i s se en t h a t the p r e s e n c e o f magnetic f i e l d a f f e c t s the t r a n s i t i o n from one t yp e to the o t h e r
t y p e o f f l o w . Fur th er, the mean v a l u e o f t h e s k i n - f r i c t i o n de cre as es as the magnetic parameter i n c r e a s e s . I t i s a l s o found t h a t an
i n c r e a s e i n the magnetic parameter r e s u l t s i n a d e c r e a s e m the v a l u e o f the 'mean d i s pl a ce m e nt t h i c k n e s s ' .
An I n t r o d u c t o r y D i s c u s s i o n o f Hy drome g ne t i c s
1, I n t r o d u c t i o n , 1
2* Hydromagnetic ^pproximci Lions and Equations o f Moti on , 6
3. Mote,tion and C o r i o ^ i s F o r ce s, 11 4. Heat T r a n s f e r i n Hydromagnetics, 13
5. E x t e r n a l and I n t e r n a l Problems i n I'y dro m ag net ics , 18
6. P r e s e n t I n v e s t i g a t i o n s , 21
An Exact S o l u t i o n i n R o t a t i n g Flow 1. I nt ro due t i o n, 27
2* .Equation o f iVotion, 28
3* Transformed Equations and t h e i r S o l u t i o n , 31
4. Conclusions, 34
E f f e c t s o f C o r i o l i s Forces on R a y l e ig h * Problem i n Hvd romsgnetics,
1* I n t r o d u c t i o n ,
36
2. Equations o f ho t i o n , 37 3. S p e c i a l Case ti, — v , 42
4. The ( V i s c o u s ) Boundsry L a y e r snd the Flow o u t s i d e , 45
5# Small and L a r g e v a l u e s o f
t
and .. e s id u a l F i e l d s , 486* Ghee r i n g s t r e s s a t the p l a t e , 54 7 . Y o r t i c i t y , 56
E f f e c t s o f S u c t i o n on R a y l e i g h Problem H yarcmegnetics
1* I n t r o d u c t i o n , 57
2. Equations o f M o ti on , 60
C h a p t e r P a g e 3. S p e c i a l ca se o f v - Tj, i * e . ,
Ma gn et ic P r a n d t l Mwiber equ als U n i t y , 64
4 . L a ^ n e t i c P r < n d t l Mimber c l o s e to one,
69
5* j>bn~conducting p l a t e , 78
. V Hydroi.itg n e t i c Flow P a s t an I n f i n i t e
83
- 110 F l a t P l a t e v.dth v a r i a b l e s u c t i o n1* Introduction, 83 2* V e l o c i t y F i e l d , 85 3. Temperature f i e l d , 98
V I Hydromagnetic F l u c t u a t i n g Flow pa st 111 — 162 an I n f i n i t e F l a t P l a t e i n the s l i p
.l e g io n
1. I n t r o d u c t i o n , 111
2* Equations o f Moti on , 114 3» S o l u t i o n o f the Problem, 117
4
* Hydromagne t i c Flow p a st a porousf l a t p l a t e i n the n o - s l i p r e g i o n , 121 5* . l e s u l t s f o r the f l o w i n the
’ s l i p ' regime, 125 6, Energy Equation, 128
7* Temperature F i e l d i n the n o - s l i p r e g i o n ,
138
8. Temperature F i e l d i n the ’ s l i p ’ regim e, 141
- V I I Hydromagne t i c Flow o f a Visco us F lu i d 163 - 183 Pa st on I n f i n i t e P l a t e w i t h ti me -de pen
dent s u c t i o n
1® I n t r o d u c t i o n , 163
2 S Equation o f L o t i o n , 164
3« P e r i o d i c v a r i a t i o n o f s u c t i o n w i t h a co ns t an t f r e e stream v e l o c i t y ,
165
4. P e r i o d i c v a r i a t i o n o f s u c t i o n w i t h ap e r i o d i c f r e e stream, 170 5* P a r t i c u l a r c a s e o f s u c t i o n i n
phtse w ith the f r e e stream, 175*
6» The c a s e o f s l o w l y changing v e l o c i t e s , 176
C h a p t e r P ag e
V I I I Hydroraagnetic F l u c t u a t i n g Flow 184 - 210 ne~r a S t a g n a t i o n P o i n t
1* I n t r o d u c t i o n , 184
2. Equations o f U o t i o n , 186 3. Exact Mimerical S o l u t i o n o f
the Equ atio n f o r the Mean Flow, 191 4* Low Frequency, 194
5. High Frequency, 195 6.
3 l : i r -
F r i c t i o n , 196Appendix 211
R e f e r e n c e s 212
